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Let $A$ be a non-empty set, and $G$ an abelian group. The set $K$ of all functions from $A$ to $G$ is an abelian group, with addition defined elementwise by $(f+g)(x)=f(x)+g(x)$ The zero element is the function that sends all elements of $A$ into $0$ of $G$ and the negative of an
element $f$ is a function defined by $(-f)(x)=-(f(x))$
When $A=\mathbb{N}$ the set of natural numbers, and $G=\mathbb{Z}$ $K$ as defined above is called the Baer-Specker group. Any element of $K$ being a function from $\mathbb{N}$ to $\mathbb{Z}$ can be expressed as an infinite sequence $( x_1,x_2,\ldots,x_n,\ldots)$ and the elementwise addition on $K$ can be realized as componentwise addition on the sequences: $$( x_1,x_2,\ldots,x_n,\ldots)+( y_1,y_2,\ldots,y_n,\ldots)=
(x_1+y_1,x_2+y_2,\ldots,x_n+y_n,\ldots).$$ An alternative characterization of the Baer-Specker group $K$ is that it can be viewed as the countably infinite direct product of copies of $\mathbb{Z}$ $$K=\mathbb{Z}^{\mathbb{N}}\cong\mathbb{Z}^{\aleph_0}= \prod_{\aleph_0}\mathbb{Z}.$$
The Baer-Specker group is an important example of a torsion-free abelian group whose rank is infinite. It is not a free abelian group, but any of its countable subgroup is free (abelian).
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- P. A. Griffith, Infinite Abelian Group Theory, The University of Chicago Press (1970)
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